Unless you have a folding bike which can be "disguised" in a suitcase, United is horrible about charging fees for bicycles. It is currently $100 each way within the US and Canada, and $200 elsewhere. Standard bikes with wheels, pedals and handlebars off, don't fit the dimensional requirements. From their website:

If you can spare the bike a few days on either end, check with FedEx or others about shipping your bike. You will need a destination (bike shop, hotel, etc) that you have checked with as well, but is is bound to be cheaper. Good luck and have fun.

Renting a bike is another option that is less hassle and most likely cheaper. I rented a bike for a race and even though it wasn't my bike (it wasn't even a nice bike since I got the last rental they had due to unforseen circumstances) I still had a great race. I was amazed at some of the nicer bikes people who had planned ahead were able to rent. Any time I have a chance to rent, I'd choose that over flying/shipping my bike.

Every airline will charge you an extra fee for bringing a bicycle. The fee varies by airline. One of the cheapest fees is Virgin America at $75. Make sure to consider that in your "total ticket cost" when getting an airline ticket.

Personally, I hate even the idea of putting my bike on an airline. The debate still rages on soft-case vs. hard-case and which is better. The cases can be expensive, too.

I think renting might be the best option, actually, if I were to do it for a race either overseas or a long way away.

For major events there are ususally bike shops that will ship the bike for you for a reasonable cost.